Download Grade 6 Algebra

ID : cn-6-Algebra [1]
Grade 6
Algebra
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Answer t he quest ions
(1)
T he length of a rectangular f ield is 5 meters more than its breadth. If its perimeter is 42 meters,
what are its length and breadth?
(2)
T he side of a regular polygon shown is a. Write the perimeter of this regular polygon using a.
(3)
Xiao Chen is 8 years younger than Park. If Park is y years old, what is the age of Xiao Chen.
(4) Ye donated 30% of his salary to charity and spent 10% of the remaining amount. He still has
¥7875 lef t. What was his total salary?
(5)
Number of schools in Dandong is 35 less than the thrice of the number of schools in Jiujiang.
If number of schools in Jiujiang is q, how many schools are there in Dandong ?
(6) T he number of days lef t in the month of June are half of the number of days already passed.
How many days are lef t in the month ?
(7) T he sum of 2 consecutive odd numbers is 40. Find the numbers.
Choose correct answer(s) f rom given choice
(8)
T he length of a rectangular hall is 3 meter less than 3 time of width. If breadth is b meters, what
is the length?
a. 3b + 3 meters
b. 3b - 3 meters
(9) T he algebraic expression f or the statement "p multiplied by itself " is,
a. p3
b. 2p
c. p2
d. p
(10) Find the sum of f ollowing polynomials
A)
-5y3 - 6xy + 2x - 10 and 18y3 - 6y
a. 13y3 - 6xy + 2x - 6y - 10
b. 13y3 - 6xy - 2x - 6y - 10
c. -13y3 - 6xy + 2x - 6y - 10
d. 13y3 - 6xy + 2x - 6y + 10
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ID : cn-6-Algebra [2]
B)
p3 + q3 + 8pq2 + 10pq + 12 and 8q3 + 12pq2 - 5
a. p3 + 9q3 + 20pq2 - 10pq + 7
b. -p3 + 9q3 + 20pq2 + 10pq + 7
c. p3 + 9q3 - 20pq2 + 10pq + 7
d. p3 + 9q3 + 20pq2 + 10pq + 7
(11) A number is 18 more than a second number. If the sum of both these numbers is 30, f ind the
f irst number.
a. 24
b. 27
c. 25
d. 23
(12) T he dimensions of rectangular f ield are 20x - 15 and 12x + 1 units respectively. T he value of x
f or which it would be square is
a. 1
b. 3
c. 2
d. none of these
(13) Write the equation which can describe relationship between p and q.
p
q
1
0.3
2
0.7
3
1.1
4
1.5
a. q = p - 0.7
b. q = 0.4p - 0.1
c. q = 0.4p + 0.1
d. q = p + 0.7
Fill in t he blanks
(14) A number if multiplied by 4 and 5 is added to the product. If result is 37, original number was
?
Check True/False
(15) An equation with single variable, where highest power is one is called linear equation.
T rue
False
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ID : cn-6-Algebra [3]
Answers
(1)
13 meters, 8 meters
Step 1
It is given that the length of the rectangular f ield is 5 meters more than it's breadth.
Let's assume the breadth of the f ield is w,
the length of the f ield will be w + 5.
Step 2
Since we know that the perimeter of a rectangle is twice the sum of length and breadth of
the rectangle.
T he perimeter of the f ield = 2(w + w + 5) = 2(2w + 5)
Step 3
It is given that the perimeter of the rectangular f ield is 42 meters.
T heref ore, 2(2w + 5) = 42
⇒ 4w + 10 = 42
⇒ 4w = 42 - 10
⇒ 4w = 32
⇒w=
32
4
⇒w=8
Step 4
T he length of the rectangular f ield = w + 5 = 8 + 5 = 13 meters
T he breadth of the rectangular f ield = w = 8 meters
(2)
5a
Step 1
T he perimeter of a polygon is equal to the sum of the lengths of all its sides.
Step 2
We know that a regular polygon is a polygon whose all sides are of equal length.
Step 3
Looking at the picture, we f ind that the given regular polygon has 5 sides.
Step 4
Since all 5 sides have equal length, perimeter = number of sides × length of each side.
=5 × a
= 5a
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ID : cn-6-Algebra [4]
(3)
y-8
Step 1
Age of Park = y.
Step 2
Age of Xiao Chen = 8 less than y
= y - 8.
(4) ¥12500
Step 1
Let's assume the total salary of the man be x
Step 2
Af ter donating 30%, percentage amount lef t with him = 100 - 30% = 70%.
T heref ore amount remaining af ter donation = (70x / 100)
Step 3
Similarly percentage amount remaing with him af ter spending 10% = 100 - 10% = 90%
T heref ore amount remaining af ter spending = (90 (70x / 100) / 100),
=
(90)(70)x
10000
Step 4
Since remaining amount is given in question to be ¥7875,
(90)(70)x
= 7875
10000
⇒x=
(7875)(10000)
(90)(70)
⇒ x = 12500
Step 5
T heref ore, his total salary was ¥12500.
(5)
3q - 35
Step 1
It is given that the number of schools in Jiujiang = q
T he thrice of the number of schools in Jiujiang = 3q
Step 2
T heref ore, 35 less than the thrice of the number of schools in Jiujiang = 3q - 35
Hence the number of schools in Dandong = 3q - 35
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(6) 10 days
Step 1
Let us assume x is the number of days that have already passed in the month of June. We
know that the number of days in the month of June is 30,
the number of days lef t in the month of June = 30 - x
Step 2
Since the number of days lef t in the month of June is half of the number of days already
passed,
we can say that 30 - x = x ×
1
2
⇒ 30 - x =
x
2
By cross multiplying both sides
2(30 - x) = x
⇒ 60 - 2x = x
⇒ 60 = x + 2x
⇒ 60 = 3x
⇒ 3x = 60
⇒x=
60
3
⇒ x = 20
Step 3
T heref ore the number of days are lef t in the month = 30 - x
= 30 - 20
= 10 days.
(7) 19, 21
Step 1
Let x be an odd number, theref ore 2 consecutive odd numbers are x, x + 2.
Step 2
T he sum of 2 consecutive odd numbers is 40.
x + x + 2 = 40
⇒ 2x + 2 = 40
⇒ 2x = 40 - 2
⇒ 2x = 38
⇒x=
38
2
⇒ x = 19
Step 3
T heref ore, the 2 consecutive odd numbers are 19, 21.
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(8)
b. 3b - 3 meters
Step 1
It is given that, the length of a rectangular hall is 3 meter less than 3 time of width.
T heref ore,
Length of the rectangular hall = 3 times of width - 3 meters
= (3 × Width of the hall) - 3 meters
Step 2
Since width is given to be b, let's replace width by b,
Length = (3 × b) - 3 meters
Length = 3b - 3 meters
(9) c. p2
T he algebraic expression f or the statement 'p multiplied by itself ' = p × p = p2
(10) A)
a. 13y3 - 6xy + 2x - 6y - 10
Step 1
In order to add the given polynomials, let us f irst place like terms in same
columns and add as usual:
-5y3 -6xy +2x
+18y3
-10
-6y
+13y3 -6xy +2x -6y -10
Remember: Using columns helps us to match like terms together in a complicated
sum.
Step 2
Sum of polynomials -5y3 - 6xy + 2x - 10 and 18y3 - 6y is 13y3 - 6xy + 2x - 6y 10.
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B)
d. p3 + 9q3 + 20pq2 + 10pq + 7
Step 1
In order to add the given polynomials, let us f irst place like terms in same
columns and add as usual:
+p3
+q3
+8pq2 +10pq +12
+8q3 +12pq2
-5
+p3 +9q3 +20pq2 +10pq +7
Remember: Using columns helps us to match like terms together in a complicated
sum.
Step 2
Sum of polynomials p3 + q3 + 8pq2 + 10pq + 12 and 8q3 + 12pq2 - 5 is p 3 + 9q 3
+ 20pq 2 + 10pq + 7.
(11) a. 24
Step 1
It is given that the number is 18 more than the second number .
Let's assume the second number is x,
the f irst number will be x + 18.
Step 2
It is also given that the sum of both these numbers is 30.
T heref ore, x + 18 + x = 30
⇒ 2x = 30 - 18
⇒x=
12
2
⇒x=6
Step 3
T he f irst number = x + 18 = 6 + 18 = 24.
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(12) c. 2
Step 1
If you look at the question caref ully, you will notice that the dimensions of rectangular f ield
are 20x - 15 and 12x + 1 units respectively.
Step 2
We know that all dimensions of a square are equal. Since the rectangular f ield would be
square, all dimensions of rectangular f ield are equal,
theref ore 20x - 15 = 12x + 1
⇒ 20x - 12x = 1 + 15
⇒ 8x = 16
⇒x=
16
8
⇒x=2
Step 3
T heref ore the value of x is 2.
(13) b. q = 0.4p - 0.1
Step 1
If you look at the all equations caref ully, you will notice that the equation q = 0.4p - 0.1
satisf y the relation between p and q.
Step 2
At all values of p, the equation "q = 0.4p - 0.1" satisf ies the corresponding values of q.
f or example q = 0.4p - 0.1
⇒ q = 0.4 × 1 -0.1
⇒ q = 0.4 -0.1
⇒ q = 0.3
the equation is satisf ies at p = 1.
Step 3
T heref ore we can say that the equation which can describe relationship between p and q is
"q = 0.4p - 0.1".
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(14)
8
Step 1
Lets x be the original number.
Step 2
If you look at the question caref ully, you will notice that if x is multiplied by 4 and 5 is added
to this product, then result is equal to 37.
or we can say that 4x + 5 = 37
⇒ 4x = 37 - 5
⇒ 4x = 32
⇒x=
32
4
⇒x=8
Step 3
T heref ore the original number is 8.
(15) T rue
Step 1
A linear equation is a f irst degree equation or we can say that the highest power of a
variable in linear equation is one. A linear equation in one variable is an euation that can be
written in the f orm ax + b = c where a, b and c are real numbers and a ≠ 0. f or example 2x +
5 = 6.
Step 2
T heref ore the statement "An equation with single variable, where highest power is one is
called linear equation." is T rue.
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